arXiv subjects

Tara Holm

Publications and source records attributed to Tara Holm.

10 recordsLinked to original sources

arXiv2025

Equivariant cohomology of a complexity-one four-manifold is determined by combinatorial data

For Hamiltonian circle actions on compact, connected, four-dimensional manifolds, we give a generators and relations description for the even part of the equivariant cohomology, as an algebra over the equivariant cohomology of a point. This description depends on combinatorial data encoded in the decorated graph of the manifold. We then give an explicit combinatorial description of all weak algebra isomorphisms. We use this description to prove that the even parts of the equivariant cohomology algebras are weakly isomorphic and the odd groups have the same ranks if and only if the labeled graphs obtained from the decorated graphs by forgetting the height and area labels are isomorphic. As a consequence, we give an example of an isomorphism of equivariant cohomology algebras that cannot be induced by an equivariant diffeomorphism of manifolds preserving a compatible almost complex structure. We also provide a soft proof that there are finitely many maximal Hamiltonian circle actions on a fixed compact, connected, four-dimensional symplectic manifold.

math.SG↗

arXiv2023

Four-periodic infinite staircases for four-dimensional polydisks

The ellipsoid embedding function of a symplectic four-manifold measures the amount by which its symplectic form must be scaled in order for it to admit an embedding of an ellipsoid of varying eccentricity. This function generalizes the Gromov width and ball packing numbers. In the one continuous family of symplectic four-manifolds that has been analyzed, one-point blowups of the complex projective plane, there is an open dense set of symplectic forms whose ellipsoid embedding functions are completely described by finitely many obstructions, while there is simultaneously a Cantor set of symplectic forms for which an infinite number of obstructions are needed. In the latter case, we say that the embedding function has an infinite staircase. In this paper we identify a new infinite staircase when the target is a four-dimensional polydisk, extending a countable family identified by Usher in 2019. Our work computes the function on infinitely many intervals and thereby indicates a method of proof for a conjecture of Usher.

math.SG↗

arXiv2016

The Morse-Bott-Kirwan condition is local

Kirwan identified a condition on a smooth function under which the usual techniques of Morse-Bott theory can be applied to this function. We prove that if a function satisfies this condition locally then it also satisfies the condition globally. As an application, we use the local normal form theorem to recover Kirwan's result that the norm-square of a momentum map satisfies Kirwan's condition.

math.SG↗

arXiv2013

The topology of toric origami manifolds

A folded symplectic form on a manifold is a closed 2-form with the mildest possible degeneracy along a hypersurface. A special class of folded symplectic manifolds are the origami symplectic manifolds, studied by Cannas da Silva, Guillemin and Pires, who classified toric origami manifolds by combinatorial origami templates. In this paper, we examine the topology of toric origami manifolds that have acyclic origami template and co-orientable folding hypersurface. We prove that the cohomology is concentrated in even degrees, and that the equivariant cohomology satisfies the GKM description. Finally we show that toric origami manifolds with co-orientable folding hypersurface provide a class of examples of Masuda and Panov's torus manifolds.

math.SG↗

arXiv2012

Equivariant cohomology for Hamiltonian torus actions on symplectic orbifolds

In this paper, we study Hamiltonian R-actions on symplectic orbifolds [M/S], where R and S are tori. We prove an injectivity theorem and generalize Tolman-Weitsman's proof of the GKM theorem in this setting. The main example is the symplectic reduction X//S of a Hamiltonian T-manifold X by a subtorus S of T. This includes the class of symplectic toric orbifolds. We define the equivariant Chen-Ruan cohomology ring and use the above results to establish a combinatorial method of computing this equivariant Chen-Ruan cohomology in terms of orbifold fixed point data.

math.SG↗

arXiv2009

Torsion and abelianization in equivariant cohomology

Let $X$ be a topological space upon which a compact connected Lie group $G$ acts. It is well-known that the equivariant cohomology $H_G^*(X;\Q)$ is isomorphic to the subalgebra of Weyl group invariants of the equivariant cohomology $H_T^*(X;\Q)$, where $T$ is a maximal torus of $G$. This relationship breaks down for coefficient rings $\k$ other than $\Q$. Instead, we prove that under a mild condition on $\k$ the algebra $H_G^*(X,\k)$ is isomorphic to the subalgebra of $H_T^*(X,\k)$ annihilated by the divided difference operators.

math.AT↗

arXiv2005

Conjugation spaces

There are classical examples of spaces X with an involution tau whose mod 2-comhomology ring resembles that of their fixed point set X^tau: there is a ring isomorphism kappa: H^2*(X) --> H^*(X^tau). Such examples include complex Grassmannians, toric manifolds, polygon spaces. In this paper, we show that the ring isomorphism kappa is part of an interesting structure in equivariant cohomology called an H^*-frame. An H^*-frame, if it exists, is natural and unique. A space with involution admitting an H^*-frame is called a conjugation space. Many examples of conjugation spaces are constructed, for instance by successive adjunctions of cells homeomorphic to a disk in C^k with the complex conjugation. A compact symplectic manifold, with an anti-symplectic involution compatible with a Hamiltonian action of a torus T, is a conjugation space, provided X^T is itself a conjugation space. This includes the co-adjoint orbits of any semi-simple compact Lie group, equipped with the Chevalley involution. We also study conjugate-equivariant complex vector bundles (`real bundles' in the sense of Atiyah) over a conjugation space and show that the isomorphism kappa maps the Chern classes onto the Stiefel-Whitney classes of the fixed bundle.

math.AT↗

arXiv2004

T-equivariant cohomology of cell complexes and the case of infinite Grassmannians

In 1998, Goresky, Kottwitz, and MacPherson showed that for certain spaces X equipped with a torus action, the T-equivariant cohomology ring of X can be described by combinatorial data obtained from its orbit decomposition. Thus, their theory transforms calculations of the equivariant topology of X to those of the combinatorics of the orbit decomposition. Since then, many authors have studied this interplay between topology and combinatorics. In this paper, we generalize the theorem of Goresky, Kottwitz, and MacPherson to the (possibly infinite-dimensional) setting where X is any equivariant cell complex with only even-dimensional cells and isolated T-fixed points, along with some additional technical hypotheses on the gluing maps. This generalization includes many new examples which have not yet been studied by GKM theory, including homogeneous spaces of a loop group LG.

math.DG↗

arXiv2002

How is a graph like a manifold?

In this article, we discuss some classical problems in combinatorics which can be solved by exploiting analogues between graph theory and the theory of manifolds. One well-known example is the McMullen conjecture, which was settled twenty years ago by Richard Stanley by interpreting certain combinatorial invariants of convex polytopes as the Betti numbers of a complex projective variety. Another example is the classical parallel redrawing problem, which turns out to be closely related to the problem of computing the second Betti number of a complex compact $(\C^*)^n$-manifold.

math.CO↗

arXiv2001

A GKM description of the equivariant cohomology ring of a homogeneous space

Let $T$ be a torus of dimension $n>1$ and $M$ a compact $T-$manifold. $M$ is a GKM manifold if the set of zero dimensional orbits in the orbit space $M/T$ is zero dimensional and the set of one dimensional orbits in $M/T$ is one dimensional. For such a manifold these sets of orbits have the structure of a labelled graph and it is known that a lot of topological information about $M$ is encoded in this graph. In this paper we prove that every compact homogeneous space $M$ of non-zero Euler characteristic is of GKM type and show that the graph associated with $M$ encodes \emph{geometric} information about $M$ as well as topological information. For example, from this graph one can detect whether $M$ admits an invariant complex structure or an invariant almost complex structure.

math.SG↗